Multimodal map and complex basin of attraction of a simple hopper
C. K. Reddy
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA R. Pratap*
Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560012, India
In this paper, we study the global dynamics of a simple passive mechanical model for hopping. The hopper is a two-mass, single-spring system constrained to move in the vertical direction共under gravity兲above a rigid ground. The hopper model and its basic dynamics including the existence of incessant hopping motions have been reported elsewhere. Here, we extend the study to investigate the global dynamics of the hopper. The global map of the hopper is multimodal. We construct an approximate analytic map near the fixed points of the map and show that the fixed points exhibit one-way stability. We also show that the map is invariant under the inversion of the mass ratio of the hopper. Next, we construct the global basin of attraction of these fixed points and show that their structure is highly complex and retains form at finer scales. This structure of the basin of attraction contains regions where the fate of an arbitrary initial condition becomes unpredictable.
I. INTRODUCTION
We consider the passive hopper model studied by Chat- terjee et al. 关1兴. The model consists of two masses m1 and m2 connected together with a spring and is allowed to move only vertically under gravity above a rigid ground共see Fig.
1兲. The collisions of the lower mass with the ground are assumed to be perfectly plastic. The motivation behind this model is to study the energetics of such a locomotion without overwhelming it with external controls. Passive systems pro- vide much insight into natural dynamics and a better starting point for controls as advocated by the works of McGeer关2兴 and subsequent researchers关3,4兴. Chatterjee et al. have stud- ied the basic dynamics and associated properties of the pas- sive hopper. The most interesting and relevant result of their investigation is the existence of energy conserving periodic motions of the hopper, termed incessant hopping in Ref.关1兴. They also show the existence of a similar motion in a jug- gling model and establish an equivalence between the two models. Schiehlen and Gao 关5兴, however, were the first to discover lossless motions of this model but their investiga- tions were cursory. Approximately simultaneously with Chatterjee et al., and independently, Hagerty关6兴investigated the existence of such a motion in the context of a bouncing eccentric cylinder and reached similar conclusions. Our work here is based on the more-available and directly applicable关1兴.
The mechanical model, shown in Fig. 1, is simple; the motions it exhibits, as we show later, are complex. That, a simple deterministic dynamical system exhibits complex be- havior 关7兴, is not surprising anymore. Yet, the dynamics we investigate here is remarkable because of its existence in several simple mechanical systems such as hoppers, jugglers 关1兴, and galloping models关6,8兴. These models are capable of persistent energy conserving motions in an overwhelmingly
dissipative phase space. These motions exhibit one-way sta- bility, a fact usually dismissed as a pathological case in mathematical theories of dynamical systems 关9兴. However, one-way stable limit cycles are known to exist in bilinear models of elastoplastic oscillators 关10兴.
If we let our mechanical intuition guide us in imagining the varied motions that this hopper can have, the plastic col- lision of the lower mass with the ground is likely to mess up our intuition for motions beyond the simple ones共such as the lower mass stopping dead with a thud on the ground while the upper mass oscillates兲. The collisions play an equally destructive role mathematically by destroying the otherwise smooth flow of the underlying dynamical system. The colli- sions introduce jumps in the state of the system, providing nonlinearity that is perhaps best modeled as discrete events.
This mixing of continuous flow and discrete events makes it a hybrid dynamical system. Although, the literature on hy- brid dynamical systems is fairly rich, much of it is motivated by mixed control strategies共see, for example, Ref. 关11兴and references therein兲. Here, we are interested in the discrete events 共impacts with the ground兲 mainly as the instants of instantaneous change in the state of the system. Several sys- tems with intermittent contacts 关12,13兴 form examples of such dynamical systems.
The simplicity of the model and the nonsmoothness intro-
*Electronic address: [email protected]
FIG. 1. The figure shows the passive hopper and the different phases during its motion共taken from Ref.关1兴兲.
duced by the collisions make maps an ideal choice for study- ing the dynamics of this system. However, we cannot obtain explicit maps as discussed in Ref. 关1兴. Chatterjee et al. con- structed the map numerically and studied its properties geo- metrically. The map turns out to be multimodal, i.e., it has many humps. Here, we construct an approximate analytical map near the fixed points 共that correspond to the incessant hopping motions兲and show that the fixed points are one-way stable as stated in Ref. 关1兴based on geometric properties of the numerical map. In addition, we show that the approxi- mate map is invariant under the mass reciprocity of the hop- per which implies that the dynamics near the fixed points is unaffected if the upper and the lower masses are inter- changed. We then proceed to construct the basin of attraction of the incessant motions and show that it has a complex structure. The complexity of the structure is evident from the numerically generated basins of attractions that retain struc- ture at finer scales. We also compute numerically the mea- sure of this basin for two selected values of the mass ratio and show that the measure is very sensitive to the mass ratio.
II. EQUATIONS OF MOTION AND LOSSLESS SOLUTIONS
The equations of motion of the system, the conditions for lossless solutions, and the general nature of solutions are described in detail in Ref.关1兴. In this section, we reproduce, in brief, the equations of motion and some of the associated dynamics from Ref. 关1兴 so that the reader can follow the main discussion of the paper from Sec. III with relative ease.
The equations of motion of the system can be written with the help of the free body diagrams shown in Fig. 1. The system behavior is best represented by separate equations for in-flight dynamics, collision transition, and dynamics during sustained contact of the lower mass with the ground. We use the following nondimensional variables for the equations of motion:
y1⬅ x1
共m2g/k兲, y2⬅ x2
共m2g/k兲, M⬅m2
m1, 共˙兲⬅d共 兲 d , where ⫽t and ⫽
冑
k(1⫹M )/m2 is the angular fre- quency of vibration in the flight phase.共a兲. Flight. The equations of motion for free flight in terms of the new variables are共see Fig. 1兲
共1⫹M兲y¨1⫽⫺M共y1⫺y2兲⫺1,
共1⫹M兲y¨2⫽共y1⫺y2兲⫺1. 共1兲 共b兲. Collision transition. When the lower mass lands on the ground, the collision causes a jump in its velocity but not in its displacement. Using ‘‘⫺/⫹’’ to denote before and after impact, the collision occurs when y2⫺⫽0 and y˙2⫺⭐0.
The impact transitions are y1⫹⫽y1⫺; y2⫹⫽y2⫺⫽0; y˙1⫹
⫽y˙1⫺; and y˙2⫹⫽0.
共c兲. Contact. During a period of sustained contact, y2
⬅0, y1⬍1, and (1⫹M )y¨1⫽⫺M y1⫺1.
共d兲. Lift-off transition. Lift-off from the ground contact occurs when the spring tension lifts the lower mass and y1
⫽1. The lift-off condition can also be met immediately at contact with no period of sustained contact 共if y1⫺⬎1). At lift-off from sustained contact there is no jump in position or velocity of either mass.
We define ␣⬅y˙1 at lift-off after a period of sustained contact.␣ is the key variable in the following analysis. The positions and velocities at the instant of lift off are
y1⫽1; y˙1⫽␣; y2⫽0; y˙2⫽0. 共2兲
These serve as initial conditions for the flight equations.
Note that all subsequent motions for all time are determined by␣at one lift-off. Thus, the dynamics can be characterized by a one-dimensional 共1D兲 map, ␣n⫹1⫽f (␣n) as seen in Sec. III.
Lossless solutions
An impact with nonzero speed would be dissipative.
Thus, for no dissipation, y˙2⫽0 at y2⫽0. For sustained loss- less motions, we also need conditions on the acceleration and jerk apart from the zero speed condition. If y¨2⬎0, contact would be immediately lost and there would be a subsequent collision with nonzero speed, and y¨2⬍0 would require prior ground penetration which is not possible physically. Thus, for lossless impact at y2⫽0, not only y˙2⫽0 but also y¨2
⫽0. Because y¨2⫽0, the ground clearance condition is de- termined by d3y2/d3. If d3y2/d3⬎0, grazing would be followed by a dissipative impact and hence there will be no subsequent sustained lossless motions. Thus, we must simul- taneously meet all of these conditions at the end of flight:
y2⫽0, y˙2⫽0, y¨2⫽0, and d3y2/d3⬍0.
All of these conditions can be simultaneously met in this model, no matter what the values of the model parameters, by adjustment of the single dynamic variable ␣ 共the lift-off speed of m1). To find these lossless solutions, we first solve the initial value problem for the flight phase 关Eqs. 共1兲 and 共2兲兴. Imposing the dissipation-free contact conditions on the solution at the end of the flight, we can solve for the time of flight in terms of␣. Then substituting this relationship in the zero acceleration condition for sustained contact, we finally get 共see Ref.关1兴for details兲
␣⫽tan␣. 共3兲
Equation 共3兲 has infinitely many solutions for ␣ 共and hence for f). These solutions for ␣ which give incessant hopping are denoted by ␣* and ⫽4.493 409, 7.725 251, 10.904 12, . . . . Higher values of ␣* correspond to more oscillations between collisions.
III. ONE-DIMENSIONAL MAP AND THE GENERAL DYNAMICS OF THE HOPPER
The dynamics of the hopper between two gradual lift-offs can be studied by constructing a map of the form
Yn⫹1⫽F共Yn兲, 共4兲 where Y⫽关y1 y˙1兴Tis the state vector of the upper mass and F:R2哫R2 maps Y from the nth gradual lift-off to the (n
⫹1)th gradual lift-off. F is defined as
F⬅FcⴰFfⴰFfⴰ•••ⴰFf, 共5兲 where Ff is the flight-impact map and Fcis the contact map.
Note that there can be more than one flight-impact maps and these correspond to immediate lift-offs without any period of steady contact. Ff and Fcare defined as
Ff共Y兲⫽
冦
y1⫹y˙y˙11⫺f⫺f⫹2M2f⫹共y˙M111⫹⫹共cosy˙MM1sinf⫺yf⫹1siny1cosf兲f兲冧
,共6兲Fc共Y兲⫽
再 冑
共1⫹M兲y˙12⫹M y11⫹12M⫹2 y1⫺2⫺M冎
, 共7兲where the time of flight f satisfies
f 2
2 ⫹y˙1sinf⫹y1cosf⫺y˙1f⫺y1⫽0. 共8兲 Since
Yn⫽
再
␣1n冎
,Eq. 共4兲is essentially a one-dimensional map of the form
␣n⫹1⫽f共␣n兲. 共9兲 The map f :R哫R cannot be written explicitly since the solution of the flight equations involves transcendental func- tions. Therefore, we construct the map numerically. A few maps corresponding to some typical values of the mass ratio M are shown in Fig. 2 where the multimodality of the map is self evident. The fixed points of this map correspond to the solutions of ␣⫽tan␣. We have plotted the map only up to the first five fixed points but the trend is clear enough not to warrant a longer map for our studies. The map by itself does not give enough information about the physical motion of the hopper with different initial conditions. We can, however, use the map to characterize the physical motion by consid- ering the set of initial conditions around a single fixed point and the unimodal map around it共a single hump兲. We could, for example, divide the region around the first nontrivial
fixed point␣*1⫽4.4934 into four segments—␣L, ␣B, ␣R, and ␣G—as shown in Fig. 3 and show a typical motion of the hopper in each segment. In␣L, the landing of the lower mass is followed by an immediate takeoff at the expense of huge loss of energy in the impact, which makes subsequent takeoffs to be increasingly difficult. The fixed point exhibits one-way stability共We prove the stability of the fixed point in Sec. IV.兲We denote the local interval of attraction as␣B. In
␣B, the dynamics eventually settles down to energy conserv- ing periodic hops. Since this incessant motion is very special, we discuss it in the following sections. On the right segment of the hump, a typical motion of the hopper in ␣R involves one or two gradual hops of the lower mass before it settles down to the ground, while the upper mass continues to os- cillate共as a single degree of freedom, undamped oscillator兲. The open segment of the map, ␣G, represents motions that have no hops at all. The motions described above are just representative motions. It turns out that ␣L and␣R are not continuous segments. They have rather complicated structure which we describe in Sec. VI. It is worth mentioning here that the nonsmooth transition of the map from ␣L to ␣B
共from a linear segment to a parabolic segment兲 is reflective of the qualitatively different motions of the hopper in the two segments. On the right side of the fixed point␣*, we have y1⫺⬍1 at impact. This condition implies that for all initial velocities of the form␣n⫽␣*⫹␦␣, there is no sudden take- off. However, on the other hand, for initial conditions to the left of the fixed point, i. e., conditions of the form ␣n⫽␣*
⫺␦␣, we have y1⫺⬎1. This condition assures that the spring has enough force to pull the lower mass immediately after impact共sudden takeoff兲, which is then followed by a second impact and further loss of energy. This motion leads to a further decrease in the value of ␣n⫹1. The one-dimensional FIG. 2. The one-dimensional map on the free variable ␣, the nondimensional velocity of the upper mass, tracked between two successive gradual hops. Different maps correspond to different val- ues of the mass ratio M.
map, therefore, curves downwards on the left side of the fixed point. The general nature of the motions of the hopper described above is repeated in similar segments of the other humps of the map, with one basic but inconsequential differ- ence that the mid-air oscillations increase in all segments as we consider humps associated with higher fixed points.
IV. STABILITY OF THE LOSSLESS MOTIONS The fixed points of the 1D map correspond to energy conserving motions of the hopper. We carry out a higher- order perturbation of the map around any fixed point ␣* in order to determine the stability of the fixed point and esti- mate the interval of attraction, if stable. An interval of attrac- tion on the left, close to the fixed point, would require the one-dimensional map to lie above the line of unit slope, which is not admissible from energy considerations. In what follows, we show that the fixed point has a nonzero interval of attraction on the right side. Consider the initial conditions at the nth gradual hop,
Yn⫽
再
␣1n冎
.The time of flight then satisfies
f 2
2 ⫹␣nsinf⫹cosf⫺␣nf⫺1⫽0, 共10兲 where ␣n⫽␣*⫹␦␣ with ␦␣Ⰶ1. Equation 共10兲 is rear- ranged as
␣n⫽
f 2
2 ⫹cosf⫺1
f⫺sinf ⫽共f兲. 共11兲 A Taylor series expansion around * 共time of flight corre- sponding to␣*) yields
␦␣⫽
f␦f⫹1 2
2
f
2共␦f兲2⫹1 6
3
f
3共␦f兲3⫹•••. 共12兲 The fixed point␣*satisfies
␣*sin共2␣*兲⫹cos共2␣*兲⫽1,
␣*cos共2␣*兲⫺sin共2␣*兲⫽⫺␣*. 共13兲 Using Eqs.共10兲and共13兲, it can be shown that
f共*兲⫽2
f
2共*兲⫽0; 3
f
3共*兲⫽ ␣*
2␣*⫺sin共2␣*兲. Neglecting the higher-order terms in ␦, we have
␦⫽共␦␣兲1/3, 共14兲 where
⫽
冋
6关2␣*⫺␣sin*共2␣*兲兴册
1/3.In Sec. III, we noted that for initial conditions of the form
␣n⫽␣*⫹␦␣, we have a period of sustained contact at land- ing. Equation共4兲, thus, is
Yn⫹1⫽F共Yn兲⫽FcⴰFf共Yn兲. Equations共6兲and共7兲give
␣n⫹1⫽
冑
共1⫹M兲共y˙1兲2⫹M1⫹共yM1兲2⫹2y1⫺2⫺M, 共15兲 wherey1⫽M共␣nsinf⫹cosf兲⫹␣nf⫹1⫺共f 2/2兲
1⫹M ,
y˙1⫽␣n⫹M共␣ncosf⫺sinf兲⫺f
1⫹M . 共16兲
Note that when␣n⫽␣*, FIG. 3. Typical dynamics of the hopper on various segments of
the map. The region around a fixed point is divided into four seg- ments:␣Lrepresents the left leg of the map,␣Brepresents the local basin of attraction of the fixed point,␣Rrepresents the right leg of the map, and ␣G represents the gap between two map segments.
Typical motions of the hopper in each segment are shown in the adjacent plots.
Ff共Yn兲⫽
再
⫺1␣*冎
.A variation␦␣1 in␣n⫹1 can be written as
␦␣1⫽␦y1
␣*⫺␦y˙1⫹ M
2␣*共1⫹M兲共␦y1兲2⫹ 1
2␣*共␦y˙1兲2⫹•••, 共17兲 where␦y1 and␦y˙1 are the variations in y1 and y˙1, respec- tively. Using Eqs. 共14兲and共16兲 in共17兲, and neglecting the higher-order terms, we get
␦␣1⬇␦␣⫺M关2␣*⫺sin共2␣*兲兴
共1⫹M兲2 共␦␣兲4/3. 共18兲 Thus, the one-dimensional map close to the fixed point␣*is given as
␣n⫹1⫽␣n⫺共␣n⫺␣*兲4/3, 共19兲 where⫽M关2␣*⫺sin(2␣*)兴/(1⫹M)2. It is clear that␣* is a fixed point and that the linearized map has a multiplier equal to 1. This condition would not guarantee stability for a general 1D map. However, in our physical system, energy dissipation prevents the map from lying above the line of unit slope. This clearly proves that the fixed point exhibits one-way stability.
A. Estimate of the local basin of attraction
The size ⌬ of the interval of attraction can be estimated by setting ␦␣1⫽0 and ␦␣⫽⌬ in Eq. 共18兲. The interval of attraction is approximately given as
⌬⫽ 1
3. 共20兲
Now, has a maximum value at M⫽1. Therefore, we ex- pect the size of the interval of attraction to be minimum at M⫽1. In the following section, we show that this is also the case for the exact map. Specifically, we show that the map is invariant when M is replaced by 1/M . Equation共19兲clearly shows the invariance for the approximate map. Figure 4 compares the approximate map and the exact map共numeri- cal兲for M⫽1.
B. Invariance of the one-dimensional map
The mass ratio M is the only system parameter in the map.
A study of the map variation with M is important for under- standing the underlying dynamics. We prove the following results for the 1D map.
Result. The one-dimensional map␣n⫹1⫽f (␣n, M ) is in- variant under M→1/M when there is no sudden jump be- tween two successive iterates of the map.
Mathematically, the above result is expressed as
␣n⫹1⫽f共␣n, M兲⫽f共␣n,1/M兲.
Using Eq. 共10兲, Eq. 共16兲is written as y1⫽␣nsinf⫹cosf
⫽a 共say兲,
y˙1⫽␣n⫺f⫹M共␣ncosf⫺sinf兲 1⫹M
⫽a1⫹M b1
1⫹M . 共21兲
Equation共15兲, thus, becomes
␣n⫹1⫽
冑
共1共⫹1⫹MM2兲兲␣2n2⫹M关a2⫹2共a⫺1兲⫹2a1b1⫺1兴共1⫹M兲2 ,
共22兲 which can be written as
␣n⫹1⫽f共␣n, M兲.
Since a, a1, b1are not functions of M, if we replace M with 1/M in Eq.共22兲, we get back the same equation. This shows that
f共␣n, M兲⫽f共␣n,1/M兲. 共23兲 It should be noted that the map is invariant only when there are no sudden lift-offs between two iterates of the map. This is true for an interval close to the fixed point on the right.
However, the entire map between two adjacent fixed points is not invariant under M→1/M .
Observation. The minimum value of the interval of attrac- tion for each fixed point occurs at M⫽1.
Although it is difficult to prove analytically that the mini- mum occurs at M⫽1 for the exact map, all numerical results suggest exactly that. From the approximate map, Eq. 共19兲, and the subsequent estimate of the basin of attraction, ⌬
⫽⫺3, it is easy to prove that the minimum size of the basin of attraction occurs at M⫽1. The invariance of the map FIG. 4. Comparison between the exact numerical map 共solid line兲and the approximate map共dotted line兲.
under the inversion of M makes M⫽1 to be an extremum point. We claim that this point corresponds to the minimum size of the basin of attraction on the basis of numerical results.
V. GLOBAL MAP AND THE EFFECT OF SYSTEM PARAMETERS
The mass ratio M is the only nondimensional parameter apart from ␣ which affects the dynamics. Figure 2 shows how the map varies with M between two successive fixed points. For low values of M, the entire map is above the line
␣n⫹1⫽␣*1. Hence, the entire interval becomes an interval of attraction. As M is increased, increasingly larger portions of the map drop below the line ␣n⫹1⫽␣*1. Consequently, the interval of attraction shrinks till M⫽1 where it is mini- mum. As M is increased further, the interval of attraction also increases. This feature leads to a complex basin of attraction as we show in Sec. VI.
VI. GLOBAL BASIN OF ATTRACTION
The local basin of attraction of a fixed point of the map is discussed in Sec. IV A. We now try to answer the following question. Are there other initial conditions apart from the ones in the local interval of attraction around a fixed point that lead to incessant hopping? That is, we seek other initial conditions, say ␣ˆ , such that fn(␣ˆ )→␣* as n→⬁. For any given M, this basin of attraction is naturally limited to 1D 关14兴. We, however, consider the variation in the mass ratio as well and construct the global basin of attraction in the M -␣ plane.
A. Construction of the basin of attraction
For a constant M, it turns out that there are infinite num- ber of intervals of initial conditions that are in the␣-limit set of a given fixed point. To illustrate this, we use a schematic drawing of the multimodal map and construct the global ba- sin of attraction of the first fixed point. The schematic map shown in Fig. 5共a兲captures the essential geometric features of the original map.
Let us consider the first fixed point␣*1. Let us denote the local basin of attraction of␣*1 by A and let S be its global
basin of attraction, i.e., S⫽兵␣兩fn(␣)→␣*1 as n→⬁其. Naturally, A傺S. Now, we consider the preimage of A. From the geometric construction shown in Fig. 5共a兲, we see that two distinct intervals A1 and A2 from the next hump map into A. That is,
f⫺1共A兲⫽A1艛A2⇒共A1艛A2兲傺S.
We can iterate backwards again and find the preimages of A1 and A2:
f⫺1共A1兲⫽A11艛A12⇒共A11艛A12兲傺S.
Continuing, thus, with the preimages, we get 艛Ai jkl . . .傺S.
Thus, the global basin of attraction S is made up of all these segments that, in the limit, form a Cantor set. The formation of the Cantor set is rather obvious from Fig. 5共b兲. However, we must point out that although S contains a Can- tor set, its topology is not equivalent to that of a Cantor set since it contains all the previous pieces of the Cantor set construction which have finite measure. Since the topologi- cal character of the map is the same for all fixed points, the structure of S remains the same for all fixed points.
B. Nested basins of attraction
We have infinite number of fixed points and every fixed point has its own basin of attraction, entirely disconnected from the basins of attraction of other fixed points. Each basin of attraction has a complex topology since it contains sets of finite as well as zero measure 共with infinite number of ele- ments兲. The global basin of attraction of lossless motions consists of the global basin of attraction of each fixed point.
Thus, we get an intricately woven, or rather nested, basin of attraction. This intricate nesting becomes more and more pronounced as we explore higher values of ␣.
The numerically generated basin of attraction is shown in Figs. 6–9. We follow a color scheme 共shown as shades of gray in the printed figures兲. We plot all points which lead to the first fixed point␣*1⫽4.493 with green, the points which lead to the second fixed point ␣*2⫽7.7252 with red, the points leading to the third fixed point with blue, and so on 共Fig. 6兲.
The figure shows that at lower values of M, the global basin of attraction is fairly large. In fact, for M⫽0.1, the entire range of ␣ shown in Fig. 6 looks green, i.e., most points seem to belong to the basin of attraction of ␣*1. To validate this, we took ␣*1⭐␣⭐␣*6, subdivided the range into five intervals 共going from ␣*i to ␣*i⫹1), took 10 000 equally spaced points in each interval and tracked the fate of their orbits. We calculated the fraction of the number of ini- tial conditions from each interval that eventually ended up in the basin of attraction of one or the other fixed point. We then added up the fractions from each interval to estimate the measure of the basin of attraction of each fixed point as well as all of them together within, of course, the finite range of
␣. For M⫽0.1, we found the measure of the ‘‘global’’ basin FIG. 5. 共a兲A schematic map showing the local basin of attrac-
tion A of the first fixed point, its preimages A1and A2, and their preimages A21and A22, etc.共b兲Construction of the basin of attrac- tion of␣*1leads to a Cantor set. The basin of attraction contains all segments including the Cantor set.
共up to␣*6) of attraction of incessant motions to be 1 Thus, all initial conditions within the selected range ended up in the basin of attraction of one or the other fixed point. A majority of these points, however, get attracted to the first fixed point whose basin of attraction measures 0.953. This is why most points in Fig. 6 look green at M⫽0.1. However, for M⫽0.2, the same measure共for all fixed points together兲 turns out to be 0.037, i.e., only 3.7% of the initial conditions lead to lossless motions. As Fig. 6 shows, this measure of the initial conditions that belong to the global basin of attraction of the incessant motions decreases rapidly as M increases and then starts increasing again as M becomes very large.
It is evident from the basins of attraction that this simple system exhibits complex dynamics. Although the basin of attraction of any one fixed point is filled with holes that
belong to the basin of attraction of other fixed points and hence seems riddled关15兴, it remains to be proved if these are indeed riddled basins关16兴. Although it is difficult to predict the fate of an arbitrary initial condition, especially for very large␣, because in the neighborhood of every initial condi- tion which would settle down to a particular fixed point, there exists another initial condition which would settle down to some other fixed point or may not settle down to any fixed point at all, the degree of unpredictability in the qualitative dynamics of the systems is not as pronounced as in the system described by Sommerer and Ott 关15兴. It is debatable whether two extremely close initial conditions leading to two different fixed points in this case qualify to be called seeds of qualitatively different dynamics. From a prac- tical point of view, both such motions are lossless incessant motions of the hopper. However, they have considerably dif- ferent energies and their mid-air dynamics are certainly quite different. Apart from such motions, any initial condition共for FIG. 6. The basin of attraction for the hopping model for small
M. The initial conditions whose iterates converge to a particular fixed point are shown by a particular shade of gray or color. The inset is shown in detail in Fig. 8.
FIG. 7. The basin of attraction for the hopping model for M
⫽1 –10. The structure in the basin for large M共close to M⫽10兲 contains similar structure as that for small M共see Fig. 6兲.
FIG. 8. The inset of Fig. 6 is shown in greater detail here re- vealing the structure of the basin. The structure at finer scales is evident in the figure.
FIG. 9. The inset of Fig. 8 is shown here revealing the structure of the basin at a further finer scale.
large␣) contains points in its neighborhood which lead to no hops at all or a few initial hops and then no hops.
VII. CONCLUSIONS AND REMARKS
We have studied the global dynamics of a passive hopper, modeled by two masses connected with a single spring. The basic dynamics of the model, including the existence of in- cessant motions, has been reported previously关1兴. Here, we have studied the multimodal map associated with the dynam- ics of the hopper in greater detail. The map variable ␣, the nondimensional velocity of the upper mass, completely char- acterizes the energy of the system at gradual takeoffs. There is only one system parameter—the mass ratio M. We use the one-dimensional map to first describe the dynamics of the hopper in different regions of the map for a given value of M and then show how the dynamics changes when we vary M.
In particular, we show that the map close to the fixed points is invariant under the inversion of M, i.e., M→1/M .
Next, we show that the fixed points exhibit one-way sta- bility by constructing and analyzing an approximate analyti-
cal map. We use the approximate map to estimate the size of the local basin of attraction. We also show that the size of the local basin depends upon the system parameter M. We then construct the global basin of attraction, the set of all initial conditions, for different values of the mass ratio M, that settle down to a fixed point. This basin of attraction is com- plex and shows structure at finer scales. The complex struc- ture of the basin contains regions with interwoven Cantor- like sets that render predicting the final state of an initial condition difficult. We have also estimated the measure of the basin of attraction over a finite range of␣for a given M.
ACKNOWLEDGMENTS
This work was supported by the Department of Science and Technology under the Robotics and Manufacturing PAC and by NSG funding. The authors wish to thank Professor Anindya Chatterjee for his suggestion on computing the measure of the basins, and Professor Andy Ruina for his critical comments on the paper and for bringing Hagerty’s work to our attention.
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